RETURN CO-MOVEMENTS AND VOLATILITY
SPILLOVERS ACROSS UNITED STATES OF
AMERICA AND EURO AREA STOCK MARKETS
Rodrigo Augusto Neves de Matos Duque
Project submitted as partial requirement for the conferral of MSc in Finance
Supervisor:
Prof. Doutor José Dias Curto, Associate Professor, ISCTE-IUL Business School, Department of Quantitative Methods
Co-supervisor:
Prof. Doutor Pedro Pires Ribeiro, Assistant Professor, ISCTE-IUL Business School, Department of Finance
ABSTRACT
We strive to analyze the interaction between stock markets of United States of America (USA) and the major countries of the euro are, by implementing a dynamic conditional correlation model to capture return co-movements and a generalized vector autoregressive model to measure volatility spillovers where forecasted-variance decompositions are independent of sorting. Impacts of recent economic crises are considered, as we analyze data from January 2000 to January 2015. Countries involved are Belgium, Finland, France, Germany, Greece, Ireland, Italy, the Netherlands, Portugal and Spain – 10 of the first 12 countries to be part of the euro area – and USA. Greece and USA appear as the two markets with lowest return co-movements with other countries, whilst France and the Netherlands show themselves as the strongest ones. As both the dotcom and the subprime crises intensify, the spillover ratio enlarges to reflect the increasing interdependency of financial markets during times of depression. Also, the sovereign debt crisis and the recent Russian financial downfall coincide with the growth of the spillovers ratio. In general, empirical results indicate high return co-movements and volatility spillovers across markets. Additionally, we assess the connection between systemic risk and volatility spillovers.
JEL classification: G10; C32
Keywords:
RESUMO
O objetivo deste trabalho é analisar a interação entre os mercados de ações dos Estados Unidos da América (EUA) e dos principais países da zona euro, através da implementação de um modelo dinâmico de correlações condicionais com o intuito de capturar possíveis relações entre retornos e um modelo generalizado de um vetor autorregressivo para calcular repercussões na volatilidade em que a decomposição das variâncias previstas é independente da ordem como estão organizadas. Os impactes das recentes crises económicas são considerados, uma vez que analisamos dados entre janeiro de 2000 e janeiro de 2015. Os países considerados são Alemanha, Bélgica, Espanha, Finlândia, França, Grécia, Holanda, Irlanda, Itália e Portugal – 10 dos primeiros 12 países que adotaram o euro como moeda única – e os EUA. Grécia e EUA aparecem como os dois mercados com menores correlações com outros países, enquanto França e Países Baixos destacam-se pela sua forte ligação com os restantes índices. À medida que as crises da dotcom e do subprime se intensificam, o rácio da difusão da volatilidade vai aumentando, o que reflete a crescente interdependência dos mercados financeiros em tempos de crise. Além disso, a crise da dívida soberana e a recente queda financeira russa coincidem também com o crescimento deste rácio. De um modo geral, os resultados empíricos demonstram fortes relações entre retornos e altos níveis de dispersão da volatilidade entre mercados. Adicionalmente, avaliamos a relação entre risco sistémico e a difusão da volatilidade.
Classificação do JEL: G10; C32
Palavras-chave:
Co-movimentos do retorno de ações, Repercussões de volatilidade, VAR generalizado, Risco sistémico
ACKNOWLEGMENTS
Just recently I read a quote from Susan Gale that truly synthesized my experience during my masters and thesis: “The more difficult it is to reach your destination, the more you’ll remember and appreciate the journey”. In retrospect, the best part was that I was never alone. Multiple persons have assisted me throughout this stage of my education and are worthy of acknowledgement. First my supervisor, Professor José Dias Curto, who has offered substantial and meaningful insight from the outset. In fact, he was the reason why I chose to address this topic in the first place. His influence and inspiration are well appreciated.
Also worthy of a special mention is Professor Pedro Pires Ribeiro for expended his time assessing my work repeatedly. His feedback and vast knowledge had a major impact over the course of this thesis, which I am grateful for.
Other people have showed their support in different ways. These include my aunt Marília, Mrs. Cândida Bento and my colleagues from ISCTE António Caetano, João Mendes, João Ramos, and Gonçalo Bernardo. To them and to all my friends and colleagues, I am thankful.
An especial acknowledgement to my girlfriend Daniela Cruz for enduring all the stress and discussing all my doubts and problems regarding this work. It is always reassuring to express our concerns and know that other people care about what is engaging your mind.
Finally, I need to thank my family, particularly my parents and my grandparents, Maria Alzira and José for supporting me and helping me get where I am today. A special thank you to my uncles José Carlos and José, and my aunts Belinha and Bélinha. Without you, this would not be possible.
ACRONYMS
BE – Belgium EUR – Euro
CISS – Composite Indicator of Systemic Stress
CISSequity – Composite Indicator of Systemic Stress for equity DCC – Dynamic Conditional Correlation
DE – Germany
D&Y – Diebold and Yilmaz
ESCS – European Coal and Steel Community ES – Spain
FI – Finland FR – France
GARCH – Generalized Autoregressive Conditional Heteroskedasticity
IGARCH – Integrated Generalized Autoregressive Conditional Heteroskedasticity GR – Greece
IE – Ireland IT – Italy
NL – Netherlands
PIIGS – Portugal, Ireland, Italy, Greece and Spain PT – Portugal
FFSI – Fed Financial Stress Index
FRED – Federal Reserve Economic Data SES – Systemic Expected Shortfall US – United States
USA – United States of America USD – United States dollar VAR – Vector Autoregressive
INDEX ABSTRACT ... I RESUMO ...II ACKNOWLEGMENTS ... III ACRONYMS ... IV INDEX ... V INDEX OF TABLES ... VI INDEX OF FIGURES ... VII INDEX OF EQUATIONS ... VIII
1. Introduction ... 1 2. Literature review ... 3 2.1. Volatility spillovers ... 3 2.2. Return co-movements ... 6 2.3. Systemic stress/risk ... 8 3. Data description... 10 4. Methodology ... 11 4.1. Return co-movements ... 12 4.2. Volatility spillovers ... 14 5. Empirical results ... 16 5.1. Descriptive statistics ... 16
5.2. Stock return co-movements ... 20
5.3. Volatility spillovers ... 26
5.3.1. Rolling sample ... 29
5.3.2. Net directional spillovers ... 31
6. Sensitive analysis ... 33
6.1. Spillover Index ... 34
6.2. Rolling Sample ... 34
7. Systemic stress and volatility spillovers ... 35
7.1. Overall and equity CISS vs spillover index ... 36
7.2. Fed Financial Stress Index vs spillover index ... 37
8. Conclusions ... 38
9. Bibliography ... 40
INDEX OF TABLES
Table I: Descriptive statistics of stock returns... 17
Table II: Tse estimates ... 21
Table III: GARCH estimates ... 22
Table IV: Step one - IGARCH estimates ... 24
Table V: Step two - correlation estimates from the standardized residuals ... 25
Table VI: Volatility spillovers of returns ... 27
Table VII: Regular volatility spillovers versus sensitive analysis ... 34
Table VIII: Unconditional correlations amidst the returns of stock markets ... 44
Table IX: Volatility spillovers of the sensitive analysis ... 44
Table X: Unconditional correlations before and after the subprime crisis between the returns of S&P 500 and Euro Stoxx 50 ... 45
INDEX OF FIGURES
Figure I: American and euro area indexes in levels ... 18
Figure IA: American and euro area indexes in levels before the subprime crisis ... 19
Figure IB: American and euro area indexes in levels after the subprime crisis ... 20
Figure II: Spillover index with a 100 weekly rolling sample ... 29
Figure IIA: Spillover index with a VAR (3), 15 ahead forecast and 150 weekly rolling sample ... 30
Figure IIB: Spillover index with a VAR (4), 20 ahead forecast and 200 weekly rolling sample ... 31
Figure III: Net spillovers with a 100 weekly rolling sample ... 31 - 32 Figure IV: Sensitive spillover index versus regular spillover index using a two year rolling sample ... 34
Figure V: Overall CISS and spillover index with a 100 weekly rolling sample ... 36
Figure VI: Equity CISS and spillover index with a 100 weekly rolling sample ... 36
INDEX OF EQUATIONS
Interpolation method for the missing data (I) ... 10
Natural logarithm for daily returns (II) ... 11
Realized weekly volatilities (III) ... 12
Dynamic conditional correlation model (IV-VIII) ... 13
Total spillover index (IX-XIII) ... 14 - 15 Directional spillovers (XIV-XV) ... 16
Net spillovers (XVI) ... 16
Unconditional correlations (XVII) ... 20
IGARCH’s symmetric positive definite matrix (XVIII) ... 24
1. Introduction
Throughout history, economic and financial disasters have taken place. Since the beginning of the 19th century, dozens of crises like, for example, the numerous panic situations involving the US, the Wall Street crash of 1929, the Black Monday of 1987 and, more recently, the dot-com bubble, the global financial crisis and the European sovereign debt slump, all have shown the major catastrophes that these events can have on modern society. However, even when everything seems to crumble, one thing remains intact: the human’s voracious ambition to attain wealth.
In order to enhance their economic growth, worldwide governments have resorted to economic integration to reach higher productivity. By being part of a large community, countries can achieve comparative advantages and economies of scale. While the latter refers to the benefits of expansion, comparative advantages are commonly used to express situations where a country can produce a given good or service at a lower cost. By having different relative costs, each country can specialize in the production of the good or service that is relatively cheaper and trade their excesses1.
Politics is also one of the main reasons for economic integration. For example, after the World War II, the need to join European countries resulted in the creation of the European Coal and Steel Community (ECSC). In addition to the development of a common market for coal and steel, this organization also helped prevent future war situations.
Although economic integration brings numerous benefits, the strong dependency amongst the member’s markets can also help propagate a given economic downturn, instead of containing it. Since it involves the coordination of fiscal and monetary policies, a stronger integration also means a lower ability for governments to adjust their own policies for their own benefit. However, the diffusion of an economic crises is not confined to a given country, community, or continent. As technologies evolve, information becomes easier to spread/diffuse. This means that even what happens in other parts of the globe – either good or bad – may have repercussions for us. The world
1 For more information on comparative advantages we refer to “On the Principles of Political Economy
has become a series of dots connected by technology and what happens in USA, does not stay in the USA.
“…Innovations in the U.S. stock market are rapidly transmitted to other markets in a
clearly recognizable pattern, whereas no single foreign market can significantly explain U.S. market movements”, Eun & Shim (1989, 243).
Even though this quote from 89 still applies, in part, to the present, recent events such as the subprime crisis, make a much stronger case to demonstrate the impact that the American economy can have on foreign countries. Despite being started in another continent, one can say that this crash helped expose the sovereign debt difficulties that Europe is facing. Therefore, it seems relevant to answer questions like: how big are these spillovers? Are they really important? Can we use them to shield us from sudden/unexpected changes in global markets? How the 2007 crisis affected these spillovers? And what about other major economic slumps?
Definitely, the answers to these questions can help us to have a deeper knowledge of how markets interact with each other, so that we can be more prepared to handle a future crisis.
Hence, this dissertation tries to add insight into return co-movements and, more importantly, volatility spillovers. The main goal is not to create a new method to measure these concepts, but to implement previous ones to different markets and/or timelines. We will analyze the results to perceive how these complex markets interact with each other, both empirically and theoretically.
Furthermore, this dissertation aims to evaluate the impact of major economic events that occurred since the beginning of the millennium on the volatility spillover index among the American stock market and some major euro area indexes.
Finally, we will cross our findings with two risk measures, namely, the Composite Indicator of Systemic Stress (CISS) provided by ECB and the Fed Financial Stress Index (FFSI) sourced from FRED, in order to detect any possible co-movements concerning these variables.
To address the return co-movements we will first test if there is a dynamic conditional correlation (DCC) using the framework developed by Tse (2000) and, in that case, use a DCC model to measure the relationship across markets. Engle (2002) argues that DCC models are suitable to explain different realities and offer sensitive empirical results. Volatility spillovers require a more complex approach. By using the D&Y (2009, 2012) methodology that generates a spillover index, we will try to evaluate how markets interact in terms of volatility spillovers. We will attempt to explain the origin of each market’s volatility and determine if a specific market is a net receiver or transmitter of volatility.
This thesis proceeds as follows. In section two we provide a literature review of the three main topics discussed, namely the volatility spillovers, return co-movements and systemic stress. Section three is intended to describe all of our data, while section four applies methodologies presented before. Empirical results are reported and interpreted in section five. Section six offers an overall sensitive analysis. We also aim to add some insight about systemic risk and volatility spillovers in section seven. Finally, section eight concludes.
2. Literature review
Since the Asian crisis during the late 90’s, many questions have been answered about markets’ interdependency. Nonetheless, many more are still waiting to be addressed. Economic incidents and major financial crises can expose some theoretical flaws, thus creating the opportunity and incentive for more knowledge concerning the interaction across sophisticated financial systems. For more details on market interdependence we refer to Forbes & Rigobon (2002).
After the 2007 subprime crisis that exhibited some of the main deficiencies of modern financial markets, the literature about return co-movements and especially volatility spillovers has grown significantly.
2.1. Volatility spillovers
There is not one single way to address volatility spillovers. As a matter of fact, multiple authors use different methodologies to measure these links [see for example, Kanas
(1998), Sola, Spagnolo, & Spagnolo (2002), Milunovich & Thorp (2006) or McMillan & Speight (2010)] and they all contributed to advertise volatility spillovers.
Although many articles have emerged, Diebold & Yilmaz (2009) stood apart from the others due to its simplicity and original way to measure volatility spillovers, where they confine all the information into one single index. An article that is then improved by the same authors in Diebold & Yilmaz (2012) using a different, yet similar, method to measure spillover effects.
While the paper from 2009 examines the return and volatility spillovers for stock markets all over the globe, the goal in D&Y (2012) is to evaluate the spillovers across the major finance markets in the United States of America – stocks, bonds, foreign exchange and commodity markets. Moreover, they utilize data from January 1999 to January 2010 to get a considerable number of observations, 2771 daily observations to be exact.
In their previous work, D&Y use a variance decomposition that is variable ordering dependent due to the Cholesky method. This means that the spillover index, which is the pillar of their whole work – and therefore the base of ours - can have different results based on how the variables are ordered. KlöBner & Wagner (2012) evaluate the 2009’s spillover measures in terms of robustness and computation and conclude that the overall spillover index is fairly robust. However, the reordering of the variables has a major impact on the spillover table results. This study also suggests methods capable of producing empirical results that are more suitable than the 2009’s spillover ratios. To overcome the variable ordering problem, D&Y propose a generalized Vector Autoregressive (VAR), a model firstly developed by Koop, Pesaran & Potter (1996) and then by Pesaran & Shin (1998).
D&Y (2009) also ignore directional spillovers and focus only on its total value. Furthermore, the article does not analyze how different assets of different countries interact and/or how distinctive assets classes within the same country share volatility spillovers. Such limitations serve as basis for other subsequent articles.
After the publication of D&Y’s index other authors resorted to this measure for their works. Yilmaz (2010) applies the D&Y (2009)’s methodology to address the degree of contagion across east Asian stock markets. The main findings reveal different fluctuations between return and volatility spillovers. Particularly, the volatility spillover index tends to rise sharply in times of economic turmoil.
In addition, Antonakakis (2012) aims to evaluate the impact of the introduction of the euro on the interaction of different exchange rates. Following the methodology applied in D&Y (2012), the author concludes that there is a significant share of volatility within these rates, but the link was stronger before the introduction of the euro. Furthermore, he advocates that significant US dollar appreciations are positively correlated with volatility spillovers. The euro (Deutsche mark) is the dominant net transmitter of volatility and the British pound is the net receiver of volatility before and after the introduction of the euro. Empirical data also shows that european markets have the highest spillover index value.
Conefrey & Cronin (2013) is also inspired by D&Y’s framework as it measures the gross and net spillover effects across euro area sovereign bond markets. Like many other articles, this investigation also detects a considerable increase of the spillover index around 2008. The authors suggest that Greece has becoming a relatively detached country from other bond markets after its second bailout in March 2012. Furthermore, a great spike in net volatility spillovers from PIIGs to the principal countries is observed at the time of the first Greece bailout.
More recently, Louzis (2013) aims to measure the interrelation of different financial markets in the euro area: money, stock, foreign exchange and bond markets in terms of return and volatility spillovers. The main findings of this paper suggest a high level of return and volatility spillovers amidst all markets. The volatility spillover index indicates that more than half of the forecast-error variance of the VAR model is explained by spillover effects. In particular, stock markets appear as the main contributors (transmitter) of these spillover effects and the bond market for the periphery countries2 as the main receiver, in exception of the period 2011-2012 where they were net transmitters. The author also concludes that the money market has a
particular role on volatility transmission to other financial markets during crisis or periods of low liquidity.
In summary, there is an extensive and recent empirical research showing a significant level of volatility spillovers across multiple asset classes, particularly when highly developed markets are involved.
2.2. Return co-movements
Return co-movements has been a popular and interesting topic in finance as investors try to understand “what makes markets move?” or “how are they related with each other?”. Although nowadays it is common to witness high correlations among most developed markets, that was not always the case. Half a century ago, worldwide stock markets experienced very low return co-movements with USA. In fact, all countries, with the exception of Canada, manifested correlations with USA equal or lower than 0,3 (Grubel, 1968). In fact, during the 70’s, articles such as Agmon (1972), Lessard (1976), Panton, Lessig & Joy (1976) and Hilliard (1979) were published addressing covariances/correlations within markets, but despite using different methodologies, the results are similar. Erstwhile, co-movements between stock markets were unexpectedly low.
Such characteristics allowed for higher diversification benefits. With low correlations, investors can spread their capital through several countries in order to mitigate the risk for a given return. The goal is to balance negative and positive performances, so that the investor is not overly exposed to sudden bursts of volatility. On the other hand, if markets move as one, diversification becomes meaningless.
Nevertheless, the trend slowly started to shift. As economies evolve, linkages among worldwide economies begin to strengthen and it was just a matter of time until research started to capture these small, but important signs of growing interdependence [e.g. Schollhammer & Sand (1985), Eun & Shim (1989), Jeon & Von Furstenberg (1990) and Koch & Koch (1991)].
Being the attractive topic that it is, market co-movements never stopped as an important subject for several researches. Since then, numeric studies indicate that developed stock markets are widely related, but less-developed countries are more advanced, Friedman
& Shachmurove (1997). Moreover, Forbes & Rigobon (2002) conclude that correlations are conditional on market volatility. The authors acknowledge this bias and argue that, when adjusted, it eliminates contagion. All markets experience significant levels of co-movements throughout the data considered.
In particular, Bekaert, Hodrick & Zhang (2009) argues that European stocks are the only ones to display an increasing correlation. Beyond that, there is no proof that the stock returns’ interdependence has increased over recent years.
So what are the main foundations for these changes? “…as most advanced economies
deregulated their capital markets, removed barriers to international investments, and improved the accessibility to information, investors in many countries have adopted a global view” (Friedman & Shachmurove, 1997, 257). Quinn & Voth (2008) also point
out capital openness as the main reason for the increasing levels of co-movements. As efforts to establish total free movement of capital continue to bring markets closer, diversification strategies are becoming less effective.
Return co-movements and volatility spillovers are closely tied together. In fact, one of the articles already mentioned in the previous sub-topic - Antonakakis (2012) - addresses both. The author argues that results show a high degree of return co-movements in some European exchange rates, but they are, on average, greater before the introduction of a single currency. Like what happens for the spillover index, US dollar appreciations appear to be related to return co-movements.
Up until the early 90’s correlations were commonly measured using constant conditional correlation models that do not account for the true time-varying nature of these interactions. It has been showed that dynamic conditional correlation models (DCC) are more suitable to explain such events3, since these models also consider time-varying variances and covariances present in each series. Antonakakis (2012) or Égert & Kočenda (2011), for example, use a DCC model proposed by Engle (2002) to measure exchange-rate returns co-movements.
However, in order to apply an appropriate model, one must know if the relationships are, indeed, dynamic. Accordingly, Tse (2000) proposes a test where the constant
correlations are considered in the null hypothesis, against an alternative where correlations are dynamic. In accordance with Forbes & Rigobon (2002), the main findings indicate a time-varying interdependence between stock markets, whereas spot-future prices and exchange rates have constant correlations.
2.3. Systemic stress/risk
Risk is a very broad concept. It can be associated with a wide variety of situations, which can go from a simple paper cut to a management of a sophisticated financial derivative. Risk management can be very useful for companies in general. It enables to identify and manage potential sources of danger, so that institutions can implement optimal strategies for each specific circumstance. As explained before, risk diversification can be one of many tools available to help in these cases, as investors try to apply their money in different assets and/or asset classes in order to mitigate their investment’s risk for a given return. To do so, the investor has to consider two risk components: the specific and the systematic risks. Although the first portion represents the risk of an individual entity or asset and, thus, can be diminished through diversification, the latter cannot. Known as the overall risk of a financial market, the systematic risk is also famous for being undiversifiable. Investors must be always aware and able to identify both components and utilize the strategies that best fit each situation.
Although they share some similarities, the systematic and the systemic risks are different. While the first is associated with the whole market segment risk, the systemic risk is used to describe the threat of a possible crash of a whole industry or financial system due to a single event. The subprime crisis is a good example of what can happen if we do not consider this kind of risk.
Rochet & Tirole (1996, 733) is one the first steps towards analyzing systemic risk. The authors define it as “the propagation of a bank's economic distress to other economic
agents linked to that bank through financial transactions”.
The recent financial crisis demonstrated the gap between theory and reality. Since then, minds all around the world strive to fill that gap. The human mind is designed to find an explanation for everything, even for what is random and unpredictable. Hendry &
Grayham (2014) argue that the models that some central banks use to predict a financial crisis would also crash in that scenario. The authors emphasize questions like: how can we know with certainty that what happen in the past will happen in the future? The future is unknown, unpredictable and full of unexpected events. Forecasting models that central banks utilize are designed using past data. Thus, they take into consideration the averages and variations that occurred in the past to predict future fluctuations. Therefore, these models will remain valid as long as the financial structure stays the same. The problem is that a financial crisis is, at itself, a structural break. That is why most models fail to predict future crises.
Even though it is hard to forecast a crash, recent papers provide important insight on systemic risk. Haldane & May (2011) alert for the consequences of the introduction of new financial tools. These instruments were created to ensure optimal returns with minimal risks, but do not consider their impact on the stability of the banking system. The authors then suggest some policy lessons with the purpose of reducing systemic risk.
Acharya, Brownlees, Engle, Farazmand & Richardson (2011) show that the Systemic Expected Shortfall (SES) can be used to measure the directional contribution that a single institution can have on the systemic risk. It also introduces a single, non-complex model of systemic risk. The article shows the SES’s capability to forecast some of the upcoming risks during the subprime crisis.
Bisias, Flood, Lo & Valavanis (2012) go for a more statistical approach, as it provides 31 different measures for systemic risk.
Hollo, Kremer & Lo Duca (2012) present the Composite Indicator of Systemic Stress (CISS), a new measure of contemporaneous stress in the financial system. It is a single measurement tool based on the structure of systemic risk. It contains a methodological novelty that allows to assign more weight to occasions, where financial stress prevails in the different markets simultaneously. This has the advantage of capturing how much the financial has diffused across the whole industry or financial system. The financial stress is more harmful in situations where the diffusion is greater. The more markets
involved, the worst it is. Further in this thesis we will tackle how the volatility spillovers and the CISS move throughout time and witness their relationship.
3. Data description
In order to proceed with this study we collect data from January 2000 to January 2015, which will give almost 4000 daily prices to work with (3915 to be exact). The reason behind such period is to give us around seven years’ worth of data before and after the bankruptcy of Lehman Brothers, which was a substantial mark of the subprime crisis. This timeline also covers other major economic events such as the dotcom bubble, the worldwide economic depression, the sovereign debt crisis and the recent Russian financial slump.
Due to different holidays and non-working days in each country, an interpolation method is used to avoid excluding more data. Depending on the number of days with missing data, we applied different versions of the same formula to fill those gaps.
As mentioned,
𝑅𝑖,𝑡 = 𝑅𝑖,0+𝑅𝑖,𝑇− 𝑅𝑖,0
𝑇 × 𝑡 (I)
where 𝑅𝑖,𝑡 represents the return of the stock market 𝑖 at moment 𝑡. 𝑇 is the number of observations until the next data available, with 𝑡 being the number of days after the last missing data. Therefore, 𝑡 = 1, … , 𝑇 − 1.
This method has the advantage of avoiding any outliers. Throughout our data sample, there are cases where a stock market stays closed for one or more days (missing data), which may cause its price to jump instead of changing gradually. With this method we mitigate those jumps.
To replicate USA (US), Belgium (BE), Finland (FI), France (FR), Germany (DE), Greece (GR), Ireland (IE), Italy (IT), the Netherlands (NL), Portugal (PT) and Spain (ES) stock markets we will use the daily prices of S&P 500, BEL 20, OMX Helsinki 25, CAC 40, DAX, Athens General, ISEQ Overall, FTSE MIB, AEX, PSI 20 and IBEX 35, respectively.
Note that we only want returns that come from variations in the price and not the exchange rate. If we use the same currency for all indexes (either all in USD or all in EUR) we will be exposed to exchange rate risk i.e., we will be including the returns (and volatilities) generated by changes in the exchange rate (and not only the price), specifically the EUR/USD rate. Thus, all euro area indexes will be priced in EUR and the S&P 500 in USD.
Weekly data will also be considered to analyze the volatility spillovers. In order to do so, we just have to focus our database on a Friday-to-Friday basis. This approach has the downfall of greatly reducing the number of observations, however it allows us to deal with daily noise that might come from using daily data.
All data mentioned before is sourced from Bloomberg.
Finally, to compare the systemic stress and the volatility spillovers we will use the CISS4 for the equity market and the overall financial system, available in the European Central Bank’s data warehouse and the FFSI, sourced from FRED. Both composites have a weekly frequency and will also contemplate the period considered for the volatility spillovers.
4. Methodology
To process the data we will use the regression analysis of time series (RATS) software that guarantees the best support for VAR models, which is the pillar of this work. The software is also designed to deal with any data frequency, which is helpful because we will require daily and weekly data to address the two main topics of this thesis, return co-movements and volatility spillovers.
After obtaining the prices of each index we will use the natural logarithm to compute the daily returns, namely:
𝑅𝑖,𝑡 = 𝑙𝑛(𝑃𝑖,𝑡) − 𝑙𝑛(𝑃𝑖,𝑡−1) = 𝑙𝑛(
𝑃𝑖,𝑡
𝑃𝑖,𝑡−1) (II)
𝑃𝑖,𝑡 is the price of the stock market 𝑖 at moment 𝑡 and 𝑡 = 1, 2, 3, … , 𝑇.
The returns will allow us to compute the weekly volatilities using a common realized volatility formula, such as:
𝜎𝑖,𝑥 = √
∑𝑡 [(𝑅𝑖,𝑡−4− 𝑅̅𝑖)2] 𝑡−4
5 (III)
where 𝜎𝑖,𝑥 is the weekly realized volatility of returns for the stock market 𝑖 at week 𝑥,
with 𝑥 = 1, 2, 3, … , 𝑋 and 𝑅̅𝑖 is the average of the last 5 (Monday-to-Friday) daily returns5.
4.1. Return co-movements
A dynamic correlation analysis will be then implemented to clarify any co-movements amongst markets. Most common correlation equations assume that the variable’s interrelation is constant through time. Nevertheless, it seems judicious to test if the interaction is, in fact, constant or if it is dynamic, i.e. if it changes as we advance in time. Tse (2000) solves this problem by proposing a test to assess if the conditional correlation is constant or dynamic:
H0: constant conditional correlation (CCC) H1: dynamic conditional correlation (DCC)
If we do not reject the null, a constant correlation measure can be used to exhibit the overall magnitude of these relationships. Nonetheless, if the null is rejected, a DCC model needs to be implemented to better represent reality. Engle (2002) offers helpful insight into DCC models and alleges that they can adapt to different situations and provide sensible empirical results. For example, the model helps to deal with numerical problems that come from the usage of a large number of variables, in which the multivariate GARCH has to estimate several parameters at the same time. Additionally, it can generate covariances, correlations and variances that change throughout time. This model estimates in two steps. First, each conditional variance is estimated through a univariate GARCH model. Then, the standardized residuals that we got from the previous procedure are used to obtain the conditional correlation matrix.
The DCC model has the following specification:
𝑟𝑡 = 𝜇𝑡(𝜃) + 𝜖𝑡, and 𝜖𝑡 ~ 𝑁(0, 𝐶𝑡) (IV)
𝜖𝑡 = 𝐶𝑡1 2⁄ 𝑣𝑡, and 𝑣𝑡 ~ 𝑁(0, 𝐼) (V)
𝐶𝑡 = 𝐷𝑡𝑀𝑡𝐷𝑡 (VI)
𝑟𝑡 = (𝑟𝑖,𝑡, … , 𝑟𝑁,𝑡)′ is a 11x1 vector of stock returns, representing all markets from
Belgium to USA. 𝜇𝑡(𝜃) = (𝜇𝑖,𝑡, … , 𝜇𝑁,𝑡)′ is the conditional mean vector for the returns.
𝐶𝑡 is the conditional covariance matrix. 𝐷𝑡 = 𝑑𝑖𝑎𝑔(𝑐𝑖𝑖,𝑡1 2⁄ , … , 𝑐𝑁𝑁,𝑡1 2⁄ )′ is a diagonal matrix
with the conditional standard deviations and 𝑐𝑖𝑖,𝑡1 2⁄ can be explained by each univariate GARCH model. 𝑀𝑡 is the 𝑡𝑥(𝑁(𝑁−1)2 ) matrix for all the time-varying conditional correlations:
𝑀𝑡 = 𝑑𝑖𝑎𝑔(𝑠𝑖𝑖,𝑡−1 2⁄ , … , 𝑠𝑁𝑁,𝑡−1 2⁄ )𝑆𝑡𝑑𝑖𝑎𝑔(𝑠𝑖𝑖,𝑡−1 2⁄ , … , 𝑠𝑁𝑁,𝑡−1 2⁄ )
or 𝜌𝑗𝑖,𝑡 = 𝜌𝑖𝑗,𝑡 = 𝑠𝑗𝑖,𝑡 √𝑠𝑗𝑗,𝑡×𝑠𝑖𝑖,𝑡
(VII)
and 𝑆𝑡 = (𝑠𝑖𝑗,𝑡) is a NxN symmetric positive definite matrix: 𝑆𝑡 = (1 − 𝛼 − 𝛽)𝑆̅ + 𝛼𝑣𝑡−1𝑣𝑡−1′ + 𝛽𝑆
𝑡−1 (VIII)
and 𝑆̅ is the NxN unconditional variance matrix of 𝑣𝑡. 𝑣𝑡 = (𝑣1,𝑡, … , 𝑣𝑁,𝑡)′ is the Nx1 vector of standardized residuals. The sum 𝛼 and 𝛽 must be lower than 1. Both parameters are equal or greater than zero (nonnegative condition).
𝛼 and 𝛽 are non-negative scalar parameters that capture the effects of previous shocks and previous dynamic conditional correlations on the current on the current dynamic conditional correlation and should satisfy the condition 𝛼 + 𝛽 < 1. As 𝑆𝑡 is conditional on the vector of standardized residuals, (VIII) is a conditional convariance matrix and 𝑆̅ is the unconditional covariance matrix of 𝜂𝑡. When 𝛼 = 𝛽 = 0, 𝑆̅ in (VIII) is equivalent
4.2. Volatility spillovers
D&Y (2009) changed the way we measure volatility spillovers by introducing, in their first work, a methodology based on a VAR framework that uses forecasted-error variance decompositions to determine the spillover index. However, it had some flaws that restricted its application. The original VAR uses a Cholesky decomposition which makes the results - variance decompositions - variable ordering dependent. This means that we need to know in advance which variables are theoretically more important (if there are any) and introduce them first. It would be better to let the data “speak by itself” and tell us which variables are more relevant.
The initial VAR framework only addressed the total spillover index and not the net directional spillovers. This can be viewed as a limitation since it does not allow to see which markets are, on average, receivers or transmitters of volatility. A directional spillover index would allow us to understand part of the market’s volatility source. D&Y (2012) implement a generalized vector autoregressive framework that is based on the research from Koop, Pesaran & Potter (1996) and Pesaran & Shin (1998). This approach creates variance decompositions that are independent of variable ordering. D&Y (2012) also introduce a directional spillover to overcome some of the limitations of their previous work. In the remainder of this section we will try to summarize and explain the D&Y’s methodology.
Consider a N-variable VAR(p) that is covariance stationary:
𝑦𝑡 = ∑𝑝 𝜃𝑖𝑦𝑡−𝑖
𝑖=1 + 𝜀𝑡 (IX)
where 𝜀𝑡 is a vector of independent and identically distributed disturbances 𝜀 ~ (0, Σ).
Given that it is covariance stationary, the moving average form of (IX) is given by:
𝑦𝑡 = ∑ 𝐴𝑖𝜀𝑡−𝑖 ∞
𝑖=1 (X)
where 𝐴𝑖 = 𝜃1𝐴𝑖−1+ 𝜃2𝐴𝑖−2+ 𝜃3𝐴𝑖−3+ ⋯ + 𝜃𝑝𝐴𝑖−𝑝 and 𝐴0 is an 𝑁 × 𝑁 indentity matrix and 𝐴𝑖 = 0 for 𝑖 < 0.
Based on this, the variance decompositions allow us to determine a spillover index using H-step-ahead forecasted error variance, due to shocks from other markets or variables, i.e. shocks from 𝑗 to 𝑖 whenever 𝑖 ≠ 𝑗. If we consider the H-step-ahead forecasted-error variance decompositions as 𝜗𝑖𝑗𝑔(𝐻) for ℎ = 1, 2, 3, … we have:
𝜗𝑖𝑗𝑔(𝐻) =𝜎𝑗𝑗 −1∑ (𝑒 𝑖′𝐴ℎ𝛴𝑒𝑗)2 𝐻−1 ℎ=0 ∑ (𝑒𝑖′𝐴 ℎ𝛴𝐴ℎ′𝑒𝑖) 𝐻−1 ℎ=0 (XI)
Σ is the variance-covariance matrix for the error vector 𝜀, 𝜎𝑗𝑗 is the standard deviation of the error for the 𝑗th equation and 𝑒𝑖 is the selection vector with one as the 𝑖th element and zeros in the rest.
By construction, the sum of each variance decomposition is not equal to one, a requirement needed to build the spillover ratio. Therefore, each variance decomposition is normalized by its row (by the sum of the cells in each column of that particular row).
𝜗̃𝑖𝑗𝑔(𝐻) = 𝜗𝑖𝑗
𝑔(𝐻)
∑𝑁 𝜗𝑖𝑗𝑔(𝐻)
𝑗=1
(XII)
This help us get what we need, ∑𝑁𝑗=1𝜗̃𝑖𝑗𝑔(𝐻)= 1 and ∑𝑁𝑖,𝑗=1𝜗̃𝑖𝑗𝑔(𝐻)= 𝑁.
If we look to the diagonal elements of the matrix as the variance that the variable “generates” for itself, the other elements of the same column can be interpreted as the variance that comes from other variables, i.e. spillovers.
Thus, the total spillover is just the sum of each off-diagonal element of the matrix divided by the sum of each element of the matrix. We multiply each spillover ratio to gives a percentage. 𝑆𝑔(𝐻) = ∑𝑁 𝜗̃𝑖𝑗𝑔(𝐻) 𝑖,𝑗=1 𝑖≠𝑗 ∑𝑁 𝜗̃𝑖𝑗𝑔(𝐻) 𝑖,𝑗=1 × 100 (XIII)
As we can see the numerator does not consider cells that belong in matrix diagonal (𝑖 ≠ 𝑗).
Now that we have all the variance decompositions that we need, we can use them to create any type of spillover that we want. By considering only the elements of one given column 𝑗 we can determine how much of 𝑖’s volatility comes from other variables. We just have to sum each element of that particular column (excluding the one in the matrix diagonal) and divided that by the sum each element of that particular column (including the one in the matrix diagonal). Accordingly, the directional volatility spillover can be written as a receiver or a transmitter denoted by 𝑆𝑖.𝑔(𝐻) and 𝑆.𝑖𝑔(𝐻), respectively.
𝑆𝑖.𝑔(𝐻) = ∑𝑁𝑗=1𝜗̃𝑖𝑗𝑔(𝐻) 𝑖≠𝑗 ∑𝑁 𝜗̃𝑖𝑗𝑔(𝐻) 𝑗=1 × 100 (XIV) 𝑆.𝑖𝑔(𝐻) = ∑𝑁 𝜗̃𝑖𝑗𝑔(𝐻) 𝑖=1 𝑖≠𝑗 ∑𝑁 𝜗̃𝑖𝑗𝑔(𝐻) 𝑖=1 × 100 (XV)
A receiver represents the volatility that a market 𝑖 receives from all other markets 𝑗, whereas a transmitter indicates the volatility that a market 𝑖 transmits to all other markets 𝑗.
The difference between these two measures will tell us if the market is predominantly a receiver or a transmitter, depending on which directional spillover is greater.
𝑆𝑖𝑔(𝐻) = 𝑆.𝑖𝑔(𝐻) − 𝑆𝑖.𝑔(𝐻) (XVI)
Note that we use a VAR with two lags (p = 2)6 and a 10-step-ahead forecasted-error, thus 𝐻 = 107
.
5. Empirical results 5.1. Descriptive statistics
To analyze the statistics for each return series we have to examine table I (see next page).
6 VAR(2).
Table I: Descriptive statistics of stock returns
*Augmented Dickey-Fuller test with no intercept nor trend was implemented to verify the existence of a unit root. Critical value at 1% is – 2,58. ** 1% significant.
a) Kwiatkowski-Phillips-Schmidt-Shin test with intercept and no trend was also implemented to verify the existence of a unit root. Critical value at 1% is 0,739.
Country BE FI FR DE GR IE IT NL PT ES US Stoxx Mean 0 0 0,0001 0,0001 -0,0005 0 -0,0002 -0,0001 -0,0002 0 0,0001 -0,0001 Median 0,0004 0,0005 0,0003 0,0008 0,0001 0,0007 0,0007 0,0005 0,0001 0,0007 0,0005 0,0001 Maximum 0,0933 0,0929 0,1059 0,108 0,1343 0,0973 0,1087 0,1003 0,102 0,1348 0,1096 0,1044 Minimum -0,0832 -0,0891 -0,0947 -0,0887 -0,1367 -0,1396 -0,086 -0,0959 -0,1038 -0,0959 -0,0947 -0,0821 Std. Dev. 0,0127 0,0153 0,0148 0,0152 0,0176 0,0139 0,0151 0,0146 0,0118 0,0149 0,0124 0,015 Skewness 0,0312 -0,1043 0,0127 -0,0303 -0,1563 -0,5993 -0,0776 -0,0951 -0,2008 0,0868 -0,1539 0,0031 Kurtosis 9,3515 6,4135 8,0382 7,6188 7,6998 11,1286 7,6922 9,6677 9,8653 8,2226 11,682 7,5742 Jarque-Bera 6578 1907 4139 3479 3617 11007 3594 7254 7711 4452 12305 3411 Probability 0 0 0 0 0 0 0 0 0 0 0 0 ADF* -58,67** -60,33** -30,96** -63,35** -56,74** -59,31** -29,43** -30,07** -57,81** -62,22** -48,55** -30,64** KPSSa) 0,123** 0,265** 0,118** 0,211** 0,142** 0,160** 0,069** 0,179** 0,122** 0,095** 0,280** 0,128** 17
€- €1,000 €2,000 €3,000 €4,000 €5,000 €6,000 $500 $1,000 $1,500 $2,000 $2,500 E uro Sto x x S& P 5 0 0
S&P 500 Euro Stoxx 50
Considering all 3914 observations, DE (along with US) and GR markets have the largest and lowest average returns, respectively. The latter reflects the consequences that GR faced due to both the sub-prime and the sovereign debt crises. As a whole, the euro area (Euro Stoxx) and the US have symmetric average returns of -0,0001 and 0,0001, respectively.
Following the same thought, GR also has the largest standard deviation. A difference of, at least, 0,2 percentage points in comparison to other markets. Curiously, Portuguese stock appears as the “safest” asset.
Since the kurtosis values are, for all series, far greater than zero, it is presumable that all series are not normal distributed. As expected, that premise is confirmed as the null for all Jarque-Bera tests is rejected.
The same happens in the Augmented Dickey-Fuller (ADF) tests as the hypothesis for the existence of a unit root is rejected in each return series considered. Each series stationarity is also supported by the KPSS unit root test. Unlike the previous test (ADF) the KPSS’s null hypothesis represents the stationarity of a given series and the results displayed are consistent with the ones gotten from the ADF test, i.e., the null is never rejected.
In order to get an overall view for the major shifts since the change of millennium, we take a small glance into the American (USD) and the euro area (EUR) indexes in levels. Fig. I: American and euro area indexes in levels
Figure I shows the evolution of the Standard and Poor's 500 index (S&P 500) and the Euro Stoxx 50 index from 2000 until 2015 in levels. Both indexes are world widely
€- €1,000 €2,000 €3,000 €4,000 €5,000 €6,000 $500 $1,000 $1,500 $2,000 2000 2001 2002 2003 2004 2005 2006 2007 2008 E uro Sto x x SP5 0 0
S&P 500 Euro Stoxx 50
accepted to characterize the American and the euro area stock markets, respectively. During that period, the American market has a steadier curve, whereas the euro area is much more volatile. Nonetheless, the two indexes seem to move in the same direction. For example, both variables manifest crashes after 2000 and 2007/08 as a result of the dotcom and the sub-prime crises, respectively. Moreover, since the beginning of 2012 that both markets have an upward slope consistent with the overall economic recovery. Note that, despite experiencing a positive relationship, variations are lower in the American market which is in accordance with the fact that the S&P 500 has a steadier curve.
If we compare figures IA and IB (below), we can interpret that in the second period (IB) the lines change closely together than the ones from the previous period. This could suggest an increase in market co-movements. In fact, the orthodox unconditional correlations given in table X reflect just that, as the interdependence strengthens after the subprime crisis – from 0,52 to 0,63. Furthermore, Sandoval & Franca (2012) and Dalkir (2009) show that in situations of high volatility, like, for instance, recession periods, correlations across markets rise. In addition, Dalkir concludes that the higher levels of co-movements remain even after the turbulence dissipates. Given that investors are risk averse, a jump in uncertainty will result in a significant number of dropouts – less demand/more supply – leading to a decrease of value. Negative returns will then originate more skepticism, keeping the circle “alive”. The same mechanism can happen for positive returns, which can help create a bubble, however, studies on financial behavior argue that investors are much more affected by losses than profits. The evolution of technologies and information flows also help investors keeping track of the worldwide economy and financial assets.
€- €1,000 €2,000 €3,000 €4,000 €5,000 $500 $1,000 $1,500 $2,000 $2,500 2008 2009 2010 2011 2012 2013 2014 2015 E uro Sto x x S& P 5 0 0
S&P 500 Euro Stoxx 50
Fig. I.B: American and euro area indexes in levels after the subprime crisis, 1829 obs
In Table VIII (see annexes) are presented unconditional correlations amidst all stock indices to give us an idea of the each relationship’s strength.
𝑐𝑜𝑟𝑟𝑖𝑗 = 𝜌𝑖𝑗 =
𝜎𝑖𝑗
𝜎𝑖𝜎𝑗 (XVII)
In majority, correlations are greater than 0,6 which shows that markets experience a considerable positive interrelationship. However, there are two special cases: GR and US. In the first situation, all correlations are lower than 50%. Despite sharing the same currency (euro) these results are somewhat of expected since GR was one of the most affected countries by the sub-prime crisis and, more recently, the sovereign debt crisis. Such events raised the level of uncertainty around this country and investors became much more skeptic. In the case of US, almost half of the correlations are also lower than 0,5.
5.2. Stock return co-movements
Here we will discuss the results obtained from applying the co-movement methodology of Engle (2002) to the US with the euro area stock markets’ returns. First, we have to test if the correlations are constant or dynamic, i.e. if the correlations are time-varying or not. Accordingly, we perform the test proposed by Tse (2000). As we can see from table II, the null hypothesis for a constant conditional correlation is rejected at the 1% significance level. This would suggest that a DCC model is more accurate to represent the reality of our data. Thus, the methodology of Engle (2002) is applied to describe such interactions.
Table II: Tse test for CC Chi-Sq (55) Sig. Level
112,81 0
According to results from the Tse test displayed in table II, we can conclude that conditional correlations are dynamic since the null is rejected at the 1% level. Thus, a DCC model is the more advisable approach to represent the proper features of this sample.
The simplest case is the GARCH (1,1), ℎ𝑡 = 𝜔 + 𝛼1𝜀𝑡−12 + 𝛽
1ℎ𝑡−1, that has become the
most popular application in modeling the time-varying conditional volatility (see, e.g., Baillie & Bollerslev, 1992 or Hansen & Lunde, 2005).
As we can see from table III (see next page) all the estimates for the coefficients 𝛼 (ARCH) and 𝛽 (GARCH) are statistically significant at a 1% level, i.e. the associated terms (lagged error and conditional variance) are all statistically relevant to explain changes in the dependent variable. Further, the ARCH estimates tend to be small (less than 0,2) and the GARCH estimates are generally high. As a result, the degree of long run persistence, 𝛼 + 𝛽, is close to one, which supports long memory processes.
Peculiarly, GR emerges as the index with the greatest long run persistence, whereas the Belgian and the Portuguese stock markets show the highest short run persistence.
Table III: GARCH estimates
Panel A: Step one - GARCH estimates
Country BE FI FR DE GR IE IT NL PT ES US Constant 0,023 0,016 0,023 0,018 0,015 0,023 0,017 0,021 0,014 0,023 0,017 (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) 𝜶 0,108 0,062 0,077 0,077 0,089 0,100 0,075 0,087 0,108 0,086 0,096 (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) β 0,876 0,930 0,911 0,913 0,912 0,889 0,919 0,900 0,887 0,905 0,893 (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) (0)
* Values between ( ) represent the probability values associated with the t-student statistic.
Panel B: DCC estimates 𝜶 0,012 (0) β 0,981 (0) 22
Truthfully, if we scrutinize the results, we detect that, for the Greek market, the sum of the estimates for 𝛼 and 𝛽 is higher than one. Consequently, it is plausible that there is a unit root in the GARCH process. This feature alters how a time series develops in the long run, thus affecting our interpretation of the results. In the presence of a non-stationary process, i.e. when ∑𝑝𝑖=1𝛽𝑖 + ∑𝑞𝑖=1𝛼𝑖 = 1, an IGARCH must be deployed. By construction, this particular version of the classic GARCH modifies how we compute the symmetric positive definite matrix8 in the DCC model:
𝑆𝑡 = 𝛼𝑣𝑡−1𝑣𝑡−1′ + 𝛽𝑆𝑡−1 (XVIII)
Nevertheless, first we need to test if the sum of 𝛼 and 𝛽 is, indeed, one. The Wald test is a parametric statistical procedure that enables us to verify the accurate value of a given coefficient based on an estimate. Hence, the two hypotheses can be expressed as:
H0: 𝛼𝑖 + 𝛽𝑖 = 1
H1: 𝛼𝑖 + 𝛽𝑖 ≠ 1
After a brief group of tests, we realized that, in general, the null hypothesis is not rejected. As a result, and consistent with a study from Engle & Bollerslev (1986), an IGARCH model is more desirable to address the existence of a unit root. Subsequently, the same authors published another article9 to support the importance of IGARCH models to manage the existence of an estimated unit root, which is common on series that utilize high frequency data.
8 See equation (VIII). 9 Bollerslev & Engle (1993)
Table IV: Step one – IGARCH estimates Panel A: IGARCH estimates
Country BE FI FR DE GR IE IT NL PT ES US Constant 0,019 0,013 0,017 0,013 0,018 0,023 0,015 0,016 0,013 0,020 0,015 (0) (0) (0) (0) (0,0001) (0) (0) (0) (0) (0) (0) 𝜶 0,117 0,069 0,086 0,084 0,089 0,113 0,081 0,097 0,112 0,095 0,107 (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) β 0,883 0,931 0,914 0,916 0,911 0,887 0,919 0,903 0,888 0,905 0,893 (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) (0)
* Values between ( ) represent the probability values associated with the t-student statistic.
Panel B: DCC estimates 𝜶 0,013 (0) 𝜷 0,982 (0) 24
Panel A of table IV plots the coefficient estimates for the IGARCH process. Again, the ARCH and GARCH estimates are all statistical significant at the 1% level. Consequently, empirical evidence shows that the market’s one lagged day shock (𝛼) and one day lagged volatility (𝛽) are relevant to explain its own volatility. By construction, the sum of each market’s 𝛼 and 𝛽 is, now, equal to one.
Further, the DCC estimates (𝛼 and 𝛽) displayed on panel B are, once again, statistically relevant to explain fluctuations on the dependent variable, i.e., the impact of past shocks on current conditional correlations (𝛼) and the impact of previous dynamic conditional correlations (𝛽) are statistically significant. This suggests that the conditional correlations are time-varying and sustains that, in this case, the DCC specification is more appropriate to address the true nature of these series. Additionally, note that the estimate 𝛼 is almost null, whilst 𝛽 is practically one, which indicates that, according to equation (XVIII), the matrix 𝑆𝑡 is almost totally explained by its one day lagged self
(𝑆𝑡−1).
Table V: Step two - correlation estimates from the standardized residuals DCC covariance/correlation matrix BE FI FR DE GR IE IT NL PT ES US BE 1,00 FI 0,69 1,00 FR 0,81 0,79 1,00 DE 0,77 0,75 0,90 1,00 GR 0,44 0,43 0,43 0,42 1,00 IE 0,61 0,61 0,65 0,61 0,40 1,00 IT 0,75 0,72 0,87 0,83 0,42 0,59 1,00 NL 0,81 0,78 0,91 0,87 0,44 0,65 0,82 1,00 PT 0,59 0,61 0,64 0,60 0,40 0,50 0,62 0,62 1,00 ES 0,74 0,72 0,86 0,81 0,43 0,59 0,84 0,80 0,66 1,00 US 0,50 0,45 0,56 0,59 0,23 0,38 0,53 0,54 0,37 0,51 1,00
In step two we estimate the conditional correlations. Unlike the ones given in table VIII, these are dynamic, thus they are more suitable to represent the time-varying component. Nevertheless the results are similar. GR stands out as the country with, on average, the lowest co-movements with other markets. US also has some of the weakest interconnections in comparison to other markets. With respect to the US, the use of a
different currency unit and the independent economic policies can justify these differences.
On the other hand, stock markets from FR and the NL have, on average, a strong correlation with other countries. In section 5.3.2. we will also realize that these two countries are net transmitters of volatility spillovers.
5.3. Volatility spillovers
In this section we show and interpret the results got from applying the methodology of D&Y (2012). Realized weekly volatilities with a Friday-to-Friday approach were used as input. D&Y (2009) propose a formula following previous literature originated in Parkinson (1980) to compute daily standard deviations using the high and low values of each stock’s price, however this approach does not allow to deal with any daily noise that might come from using day-to-day data. In accordance, Black (1986) argues that noise is everywhere and it is what makes trading in stock markets possible. Traders see these fluctuations in prices as random changes (or noise) and try to profit from them by the use of algorithms. These strategies can be implemented without any use of fundamental information.
After a more wide analysis, a rolling sample will be implemented to observe the behavior of all variables throughout the period considered since it seems unlikely that any single fixed-parameter model would apply over the entire sample. In this vein, we estimate the D&Y spillover measure on a moving window basis as it allows us to construct time-varying spillover indices that reflect the economic and financial evolution.
Finally, results for net directional spillovers are displayed to conclude about which markets are net transmitters/receivers.
Table VI: Volatility spillovers of returns From (j) To (i) BE FI FR DE GR IE IT NL PT ES US From Others BE 17,5 6,2 12 8,9 2,8 7,8 8,7 13,3 6,2 9,5 7,1 83 FI 8 20 10,8 8,2 3,3 5,8 8,7 9,4 7,1 9,9 8,8 80 FR 10,9 7,5 14,8 10,9 2,4 5,3 9,7 13,7 5,9 11,2 7,5 85 DE 10,5 7,2 13,6 16,2 2 4,4 9,3 14,4 4,7 9,9 7,9 84 GR 6,3 7,1 6,6 4,2 37,8 5,3 8,1 5,3 9,3 7,2 2,9 62 IE 11,9 7,4 8,2 4,6 3,2 25,2 6 8,9 7,9 8,1 8,5 75 IT 9,9 7,4 12 9,6 3,6 4,7 16 10,4 8,1 11,7 6,7 84 NL 12,3 6,4 13,5 10,9 1,9 6,3 8 18,1 4,9 9,9 7,8 82 PT 9,3 7,9 9,3 6 5 6,6 9,4 7,6 23 10,7 5,1 77 ES 9,6 7,4 12,4 8,8 3,6 5,2 10,6 10,9 8,2 16,7 6,6 83 US 10 7,6 9,8 6,6 1,7 9,2 6,8 10,9 5,7 8,1 23,5 76 Contrib. to others 99 72 108 79 29 61 85 105 68 96 69 871 Contrib. incl. own 116 92 123 95 67 86 101 123 91 113 92 79,20% Net Spillover 16 -8 23 -5 -33 -14 1 23 -9 13 -7
Table VI shows the overall spillover index and the directional spillovers for this sample. Is important to explain how it should be interpreted to avoid any problems.
Each column represents the volatility spillover that the country j transmits to each one of the remaining markets. Therefore, the sum of all the cells, excluding the main diagonal, in a given column represents the contribution that a given country is transmitting to others. If we include the main diagonal, we are looking to the contribution including the market itself.
Following the same rationale, each row represents the volatility spillover that a given country i receives from each one of the remaining markets and its sum, excluding the main diagonal, will tells us how much spillovers a given market is receiving from all the other indexes.
The sum of all contributions to others and the sum of all from others is the same and, when divided by the sum of all contribution including the market itself, results in the overall spillover index: 79,2%.
The net spillovers are just the difference between the contribution to other and the contribution from others.
Now that we covered the interpretation of table VI, we will evaluate the obtained results. First and foremost, it is possible to see that the overall spillover index is quite high – almost 80%. This indicates that the spillovers from the market itself can only explain a small part of the share of volatility. Such a high spillover index reflects the great interdependency in today’s developed markets. This value if boosted due to the fact that almost all of our sample is composed by euro area countries: 10 out of 11. The use of a single currency helps union markets. The countries’ inability to freely adapt their economic and fiscal policies and the use of a common monetary policy make euro area countries closely linked. A shock in euro has an impact on all of them. In comparison to the D&Y (2009) results for worldwide stock markets, our spillover index is twice as big, 79,2% against 39,5%, pointing to the greater interdependency of euro area markets. This can also be explained by the fact that the sample period and, of course, the countries analyzed are different than the ones used by D&Y (2009).
In addition, GR stands out as the major receiver of spillover volatility. Since the recent crises and the struggle to produce economic growth, the fragilities of the Greek economy were exposed. Thus, any sudden changes in euro area markets would greatly impact Greek markets. The same argument can, in some degree, be made for the fact that IE and PT are also receivers of spillover volatility, as both countries faced their own harsh times.
In contrast, FR and the NL appear as the main transmitters of spillover volatility.
Although our findings appear to be quite good, they are not new in Europe. Some of the articles already mentioned here and others such as Alter & Beyer (2014), Antonakakis & Vergos (2013) and Sosvilla-Rivero & Morales-Zumaquero (2012) also show significant volatility spillovers for different financial markets in the eurozone.
5.3.1. Rolling sample
In general, table VI offers insightful information regarding overall and directional spillovers across markets. However, it does not allow to witness the fluctuations over time. In fact, it seems reckless to think that any single fixed-parameter model would apply through the entire sample. To circumvent this, a moving window basis of the overall D&Y’s spillover measure is implemented in order to construct a time-varying spillover index. Along these lines, we use a 100-week rolling sample to produce a rolling spillover ratio.
Fig. II: Spillover index with a 100 weekly rolling sample
Fig. II illustrates the evolution of the spillover index using a roll-over approach with 100 observations. This allows us to witness how diffusion patterns evolve throughout time.
It is relatively easy to notice different trends. For example, from 2002 until 2005 the spillover index increases coinciding with the post dot-com crisis, but since then until midst 2006 it starts to fall. Then, after achieving its lowest level, the index erupts until 2008/09, where it flattens until 2010, reflecting the first symptoms and developments of the subprime crisis. Notice that, during the end of that decade which was a period also marked by the Irish banking crisis, the spillover index reaches its highest level. Curiously, after that point the measure starts falling, in exception for 2011 with the emergence of European sovereign debt crisis. It is only after 2014 that the spillovers begin again to rise, at the same time that a financial slump begins to develop in Russia
60 65 70 75 80 85 90 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 (%) Spillover
(RU). Although RU is not part of our database, the overall distrust can spread out to the euro zone and possibly increases the spillover ratio.
To examine the rolling windows’ robustness, we re-computed the rolling spillover index using different magnitudes for the associated parameters. This will allow us to access if this procedure is robust to changes in: (i) the length of the rolling windows, (ii) the number of steps ahead utilized to generate the variance decomposition and (iii) the number of lags in the VAR. Formerly, we computed the rolling windows using a two lagged VAR with a 10-step-ahead forecasted-error variances and a 100 weekly rolling sample. Alternatively, we provide two different situations to infer about the overall robustness of this procedure:
A. VAR (3), 15 ahead forecast and 150 weekly rolling sample B. VAR (4), 20 ahead forecast and 200 weekly rolling sample
As we can perceive below by figures IIA and IIB, the two alternative scenarios produce similar evolutions when compared to our benchmark spillover index, thus suggesting an overall robustness of our rolling windows.
It is also possible to notice that, even though the trends and fluctuations appear to match in all scenarios, the variations are, in general, lower for the alternative settings.
Fig. IIA: Spillover index with a VAR (3), 15 ahead forecast and 150 weekly rolling sample 60 65 70 75 80 85 90 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 (%) Spillovers
